Tomographic reconstruction is a well-known technique underlying nearly all of the diagnostic imaging modalities including x-ray computed tomography (CT), positron emission tomography (PET), singly photon emission tomography (SPECT), and certain acquisition methods for magnetic resonance imaging (MRI). It also finds application in manufacturing for nondestructive evaluation (NDE), for security scanning, in synthetic aperture radar (SAR), radio astronomy, geophysics and other areas.
There are several main formats of tomographic data: (i) parallel-beam, in which the line-integrals are performed along sets of parallel lines; (ii) divergent-beam, in which the line-integrals are performed along sets of lines that diverge as a fan or a cone; and (iii) curved, in which the integrals are performed along sets of curves, such as circles, ellipses, or other closed or open curves. One problem of tomographic reconstruction is to reconstruct a 2D or 3D image from a set of its line-integral projections. Another problem of tomographic reconstruction is to reconstruct a 3D image from a set of its surface-integral projections, that is, its integrals on a family of surfaces. For example, the 3D Radon transform involves integrals of the image on a family of 2D planes of various orientations and distances from the origin. Some of the problems of tomographic reconstruction, and some of the reconstruction methods, are described in standard references such as F. Natterer, The Mathematics of Computerized Tomography. Chichester: John Wiley, 1986; F. Natterer and F. Wubbeling, Mathematical Methods in Image Reconstruction. Philadelphia: Society for Industrial and Applied Mathematics, 2001; A. C. Kak and M. Soaney, Principles of Computerized Tomographic Imaging. New York: IEEE Press, 1988; and S. R. Deans, The Radon Transform and Some of Its Applications. New York: Wiley, 1983.
The method of choice for tomographic reconstruction is filtered backprojection (FBP) or convolution backprojection (CBP), which use an unweighted (in the parallel-beam or Radon Transform cases) or a weighted (in most other cases) backprojection step. This step is the computational bottleneck in the technique, with computational requirements scaling as N3 for an N×N-pixel image in 2D, and at least as N4 for an N×N×N-voxel image in 3D. Thus, doubling the image resolution from N to 2N results in roughly an 8-fold (or 16-fold, in 3D) increase in computation. While computers have become much faster, with the advent of new technologies capable of collecting ever larger quantities of data in real time (e.g., cardiac imaging with multi-row detectors, interventional imaging), and the proliferation of 3D acquisition geometries, there is a growing need for fast reconstruction techniques. Fast reconstruction can either speed up the image formation process, reduce the cost of a special-purpose image reconstruction computer, or both.
The dual operation of backprojection is reprojection, which is the process of computing the projections of an electronically stored image. This process, too, plays a fundamental role in tomographic reconstruction. A combination of backprojection and reprojection can also be used to construct fast reconstruction algorithms for the long object problem in the helical cone-beam geometry, which is key to practical 3D imaging of human subjects. Furthermore, in various applications it is advantageous or even necessary to use iterative reconstruction algorithms, in which both backprojection and reprojection steps are performed several times for the reconstruction of a single image. Speeding up the backprojection and reprojection steps will determine the economic feasibility of such iterative methods.
Several methods have been proposed over the years to speed up reconstruction. For example, Brandt et al. U.S. Pat. No. 5,778,038 describes a method for 2D parallel-beam tomography using a multilevel decomposition, producing at each stage an image covering the entire field-of-view, with increasing resolution. Nillson et al. U.S. Pat. No. 6,151,377 disclose other hierarchical backprojection methods. While these systems may have merit, there is still a need for methods and apparatus that produce more accurate images, and offer more flexibility between accuracy and speed.
Accordingly, one object of this invention is to provide new and improved methods and apparatus for computed tomography (CT) scanning.
Another object is to provide new and improved methods and apparatus for CT scanning that produce more accurate images, and offer more flexibility between accuracy and speed.